Circular jacobi ensembles and deformed verblunsky coefficients

Abstract

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analog of the Jacobi ensemble: (Image Omitted) with ℛeδ > -1/2. If e is a cyclic vector for a unitary n × n matrix U, the spectral measure of the pair (U, e) is well parameterized by its Verblunsky coefficients (α0, ..., αn-1). We introduce here a deformation (γ0, ..., γn-1) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r(γ0)⋯ r(γn-1) of elementary reflections parameterized by these coefficients. If γ0, ..., γn-1 are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above.These deformed Verblunsky coefficients also allow us to prove that, in the regime δ = δ(n) with δ(n)/n → βd2, the spectral measure and the empirical spectral distribution weakly converge to an explicit nontrivial probability measure supported by an arc of the unit circle. We also prove the large deviations for the empirical spectral distribution. © The Author 2009. Published by Oxford University Press. All rights reserved.